Question 1209365
(1) Let a_1, a_2, a_3 be real numbers such that
|a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_1| = 1.
What is the largest possible value of |a_1 - a_2|?


(2) Let a_1, a_2, a_3, \dots, a_{10} be real numbers such that
|a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_4| + 4 |a_4 - a_5 | + \dots + 9 |a_9 - a_{10}| + 10 |a_{10} - a_1| = 1.
What is the largest possible value of |a_1 - a_6|?
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I will solve here part (a), ONLY.



<pre>
        Take  a_3 = a_1. 

         We always can do it.



Then in the equation

    |a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_1| = 1


the addend  3|a_3 - a_1|  will be zero,  the addend  2|a_2 - a_3| will become  2|a_2 - a_1|,
and the equation will take the form

    3|a_1 - a_2| = 1.


It says that value of  |a_1 - a_2|  is  1/3  then.


If to think 1 minute, placing mentally  a_3  inside the interval  [a_1,a_2] of the length 1/3
or outside such an interval, it become clear that such solution gives the optimal choice 
and produces the maximum possible value of |a_1 - a_2|.


<U>ANSWER</U>.  The largest possible value of |a_1 - a_2|  is  1/3.
</pre>

Solved.